---
tag: G0JDQQRTGV
kind: definition
name: Frattini subgroup
msc: [20D25, "20E28"]
defines: KXJG5VV340
uses: [G5S1M9, 0X1JXT47GA, 0839XNQXDK, 0RP7VWQ0YN, 0KABPMRHM2, 0Q7MMQHC1P, 06QYRTX9KQ, 0NCJ5WXN6Z, 0QTZDKJEDV]
sources:
  - url: https://en.wikipedia.org/wiki/Frattini_subgroup
---

## Definition

Let $G$ be a [[group|G5S1M9]].
A *maximal subgroup* of $G$ is a [[subgroup|0X1JXT47GA]] $M \ne G$ such that no subgroup $H$ satisfies $M < H < G$.
The *Frattini subgroup* $\Phi(G)$ is the [[intersection|0839XNQXDK]] of all maximal subgroups of $G$, with $\Phi(G) = G$ when $G$ has no maximal subgroup.

## Remarks

An element $g \in G$ is a *non-generator* if, for every subset $X \subseteq G$ with $\langle X, g \rangle = G$, already $\langle X \rangle = G$.
Then $\Phi(G)$ is the set of non-generators of $G$.
$\Phi(G)$ is a [[characteristic subgroup|0KABPMRHM2]] of $G$, and if $G$ is finite it is [[nilpotent|0Q7MMQHC1P]].
For a finite [[$p$-group|06QYRTX9KQ]] $P$, $\Phi(P)$ is generated by the $p$-th powers and the commutators, and $P/\Phi(P)$ is an elementary abelian group.

Example: $G = \Z/8\Z$.
Its subgroups form a chain $\set{0} < \set{0, 4} < \set{0, 2, 4, 6} < G$, so the only maximal subgroup is $\set{0, 2, 4, 6}$, and this is $\Phi(G)$.

Example: $G = S_3$.
The maximal subgroups are $A_3$ and the three subgroups of order $2$; their intersection is trivial, so $\Phi(S_3) = \set{e}$.

Example: the group $\Q$ of [[rational numbers|0NCJ5WXN6Z]] under addition has no maximal subgroup, so $\Phi(\Q) = \Q$.

Non-example: in $\Z/8\Z$, the subgroup $\set{0, 4}$ is not the Frattini subgroup, since it is smaller than the intersection of the maximal subgroups.

Do not confuse $\Phi(G)$ with the [[Frattini argument|0QTZDKJEDV]], which is a different result named after the same mathematician.